Particle Swarm Optimization for Simultaneous

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tolerance allocation of clutch assembly consisting of three components. This paper .... tion of the concurrent design, the cylinder-piston assembly [2] is described.
Particle Swarm Optimization for Simultaneous Optimization of Design and Machining Tolerances* Chi Zhou, Liang Gao, Hai-Bing Gao, and Kun Zan Department of Industrial & Manufacturing System Engineering Huazhong Univ. of Sci. & Tech., Wuhan, 430074, China [email protected]

Abstract. Tolerance assignment is an important issue in product design and manufacturing. However, this problem is commonly formulated as nonlinear, multi-variable and high constrained model. Most of the heuristics for this problem are based on penalty function strategy which unfortunately suffers from inherent drawbacks. To overcome these drawbacks, this paper presented a new powerful tool-Particle Swarm Optimization algorithm (PSO) and meanwhile proposed a sophisticated constraints handling scheme suitable for the optimization mechanism of PSO. An example involving simultaneously assigning both design and machining tolerances based on optimum total machining cost is employed to demonstrate the efficiency and effectiveness of the proposed approach. The experimental result based on the comparison between PSO and GA show that the new PSO model is a powerful tool.

1 Introduction Tolerance assignment in product design and process planning (machining) affects both the quality and the cost of the overall product cycle. It is a crucial issue to determine how much the tolerance should be relaxed during the assignment process, since a tight tolerance implies a high manufacturing cost and a loose tolerance results in low manufacturing cost. Hence, during tolerance assignment, a balance between a reduction in quality loss and a reduction in manufacturing cost must be considered. Traditionally, in the two stages (product design and process planning) tolerances [1] are often conducted separately. This is probably due to the fact that they deal with different type of tolerances. Product design is concerned with related component tolerances, whereas process designing focus on the process tolerance according to the process specification. However, this separated approach in tolerance design always suffers from several drawbacks. Therefore, we need to develop a simultaneous tolerance design. Singh [2] utilized genetic algorithms and penalty function approach to solve the problem of simultaneous selection of design and manufacturing tolerances based on the minimization of the total manufacturing cost. Gao and Huang [3] utilized a nonlinear programming model for optimal process tolerance simultaneously based on the objective of total manufacturing cost with different weighting factors. Huang, *

This paper is supported by the National Basic Research Program of China (973 Program), No.2004CB719405 and the National Natural Science Foundation of China, No. 50305008.

T.-D. Wang et al. (Eds.): SEAL 2006, LNCS 4247, pp. 873 – 880, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Zhong and Xu [4] proposed a robust optimum tolerance design method in a concurrent environment to balance the conflict design targets between manufacturing tolerances and product satisfaction. Doubtlessly, the tremendous achievement has been obtained in the simultaneous tolerance optimization in the concurrent engineering context. However, this problem is characterized by nonlinear objective, multiple independent variables. Traditional operational research algorithms are successful in locating the optimal solution, but they are usually problem dependent and lack of generality. Some modern heuristic methods are relatively more robust and flexible to solve these complex problems, but they may risk being trapped to a local optimum and are usually slow in convergence and require heavy computational cost. In view of the above problems and the past successful applications of PSO in nonlinear optimization, maybe PSO is a potential remedy to these drawbacks. PSO is a novel population based heuristic, which utilizes the swarm intelligence generated by the cooperation and competition between the particles in a swarm [5][6]. Noorul Hap, et al [7] utilized PSO to achieve the multiple objective of minimum quality loss function and manufacturing cost for the machining tolerance allocation of the over running clutch assembly. The presented method outperforms other methods such as GP and GA, but it considered only two dimensional tolerance allocation of clutch assembly consisting of three components. This paper attempts to solve more complex tolerance assignment problems by PSO with a sophisticated constraints handling strategy. This paper is organized as follows. In section 2, the problem of simultaneous design was described. The basic PSO algorithm was reviewed and the new sophisticated constraints handling strategy corresponding to PSO was presented in Section 3. Section 4 gave an example and the evaluation of the proposed technique is carried out on the example. Some conclusions and further discussion are offered in Section 5.

2 Simultaneous Design As mentioned before, design processes are commonly divided into two main stages: product design and process design. Dimensional tolerance analysis is very important in both product and process design. In product design stage, the functional and assembly tolerances should be appropriately distributed among the constituent dimensions, this kind of tolerances are called design tolerances. In the meantime, each design tolerance for the single dimension should be subsequently refined to satisfy the requirement for process plans in machining a part. Such tolerances for the specified machining operation are called manufacturing tolerance. However, the traditional process of design and machining tolerance allocations based on experiences can not guarantee optimum tolerance for minimum production cost. This work aimed at selecting the optimal tolerances sequences to achieve the minimum manufacturing cost considering the two types of tolerances simultaneously by a powerful global optimization tool. This problem is formulated as follows.

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2.1 Objective Function We take the manufacturing cost as the objective function. Generally, the processing of mechanical product is conducted in a series of process plans. Different process consumes different expense because different process is associated with different machining method. Therefore, the cost of manufacture of the product is the summation of all operation cost. In this work, a modified form of the exponential cost [2] function will be adopted. The manufacturing cost of the machining tolerance is formulated as equation (1).

cij (δ ij ) = a0 e − a1 (δ − a2 ) + a3

i = 1," , n

(1)

The total manufacturing cost of a product will be C, where: n

mi

C = ∑∑ cij

(2)

i =1 j =1

Where cij (δ ij ) and δ ij is the manufacturing cost and the tolerance of the jth manufacturing operation associated with the ith dimension respectively. n is the number of the dimensions and mi is number of operations corresponding to dimension i. The constants a0, a1, a2, a3 sever as control parameters. 2.2 Constraints

Apart from the constraint of economical manufacturing ranges (process limits), the above objective is subjected to both the design and manufacturing tolerances. (1) The design tolerances are those on the principal design dimensions (usually assembly dimensions) that relate to the functionality of the components. The principal design usually in turn relies on the other related dimensions which form a dimension chain. This results in a set of constraints on the principal design tolerances that should be suit for the optimal solution of the tolerance assignment. There are many approaches available to formulate the synthesized tolerance. They are different tradeoff between the tolerances and the manufacturing cost. Four commonly used approaches [2] were adopted in this work. (2) Manufacturing tolerances constraints are equivalent to stock allowance constraints. Stock allowance is associated with the stock removal, the layer to be removed from the surface in the machining process. Due to the tolerances of the dimensions, the stock removal is also not fixed. This gives rise to another kind of tolerances, manufacturing tolerances, which can be formulated as follows:

δ ij + δ i ( j −1) ≤ ΔAij

(3)

where δ ij and δ i ( j −1) are the machining tolerances of process j and j-1 for part i respectively. ΔAij is the difference between the nominal and the minimum machining allowances for machining process j.

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3 Particle Swarm Optimization 3.1 Background

The investigation and analysis on the biologic colony demonstrated that intelligence generated from complex activities such as cooperation and competition among individuals can provide efficient solutions for specific optimization problems [8]. Inspired by the social behavior of animals such as fish schooling and bird flocking, Kennedy and Eberhart designed the Particle Swarm Optimization (PSO) in 1995 [9]. This method is a kind of evolutionary computing technology based on swarm intelligence. The basic idea of bird flocking can be depicted as follows: In a bird colony, each bird looks for its own food and in the meantime they cooperate with each other by sharing information among them. Therefore, each bird will explore next promising area by its own experience and experience from the others. Due to these attractive characteristics, i.e. memory and cooperation, PSO is widely applied in many research area and real-world engineering fields as a powerful optimization tool. 3.2 Drawbacks of Traditional Constraints Handling Strategy

Although PSO has successfully solved many research problems, the applications are mainly focused on unconstrained optimization problems. Some researchers attempt to solve the constrained problem by optimizing constrained problems indirectly using the traditional penalty function strategy. Penalty function is an effective auxiliary tool to deal with simple constrained problems and has been the most popular approach because of their simplicity and ease of implementation. Nevertheless, since the penalty function approach is generic, their performance is not always satisfactory. When combined with PSO, the above problem is more severe in that PSO has an inherent mechanism based on memory information. This mechanism can produce high efficiency and effectiveness, but also low the flexibility for constrained optimization simultaneously. It is desirable to design a new constraint handling scheme suit for PSO to effectively solve numerous engineering problems and maintain high efficiency. 3.3 Constraints Handling Strategy for PSO

Taking account of the memory mechanism of PSO and penalty strategy, a new constraint-handling strategy is presented in Figure.1. The core characteristics of the proposed strategy can be described as follows: (1) Corresponding to the memory mechanism of PSO, a special notation-Particle has been Feasible (PF) is introduced, which is used to record whether the current particle has ever satisfied all the constraint conditions. This notation preserves historical constrain status for each particle. (2) Each particle updates its individual best and neighborhood best according to the historical constraint information PF, the current constrain status (Current particle is Feasible, CF) and the objective function with the penalty term.

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(3) The algorithm selects the velocity updating strategy according to the historical information PF. (4) When updating the personal and neighborhood best, the algorithm adopts the static penalty strategy instead of the dynamic and the adaptive ones to guarantee the fairness. The detailed procedure for updating the personal and neighborhood best values based on the above constrain handling strategy is presented in Figure.1.



For Each Particle { If PF true Then If f ( xi ) ≤ f ( pi ) and CF= true Then

pi

=xi

If f ( pi ) ≤ f (li ) Then

pi

=l

i

End if End if Else if PF false Then If CF true Then

= = p = xi PF=true i

If f ( pi ) ≤ f (li ) Then

pi

=l

i

End if Else if f ( xi ) ≤ f ( pi ) Then

pi

=xi

End if End if

Fig. 1. The proposed constraint handling strategy for PSO

Special attention should be paid that the PSO algorithm based on the proposed constraint handling strategy does not have to guarantee the existence of feasible solutions in the initial population. With the randomized initial velocity, the PSO itself has the ability to explore the feasible space. In addition, the penalty function imposed on the violated particles also direct the search of PSO towards the feasible region. According to the velocity updating formula, each particle will obtain updating information from its neighborhood best particle, so the corresponding particle would return to the feasible solution space immediately.

4 Design Example To validate the effectiveness of the new proposed strategy and illustrate the application of the concurrent design, the cylinder-piston assembly [2] is described. In this

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example, the piston diameter is 50.8mm, the cylinder bore diameter is 50.856mm, and the clearance is 0.056 ± 0.025 mm. The ranges of the principal machining tolerances for the piston and cylinder bore were the same as in the [2]. In this problem, we have to consider totally 10 tolerances as follows. (1)The design tolerance parameters: δ11d for the piston and δ 21d for the cylinder bore. Four stack-up conditions [2] (worst case, RSS, Spotts’ modified method and estimated mean shift criteria) are employed to formulate the corresponding constraints. (2)The machining tolerance parameters are: δ ij where i=1,2 and j=1,2,3,4. Usually, the process tolerance for the final finishing operation is same as the design tolerance, i.e. δ11d = δ14 and δ12 d = δ 24 . The machining tolerance constraints are formulated based on Equation 3. The manufacturing decision is the total machining cost and is determined by summing the machining cost-tolerance model as Equation 1 and Equation 2 subjecting to the constraints and ranges of the principal design and machining tolerances. The constant parameters are the same as in [2]. The proposed PSO algorithm with special constraints handling strategy was used to solve this problem. To validate its efficiency, this new approach was compared with GA in [2]. In the optimization process of HPSO, we set the population size popsize=80, the maximum iteration number itermax=600. These two parameters are the same as those in GA. The other parameters are set as the common used method. The inertial weight decreases from 0.9 to 0.4 linearly and the accelerated parameters c1=c2=2.

Fig. 2. Variation of the minimum, maximum and average of the manufacturing costs with progress of the algorithm (Greenwood and Chase's unified, or estimated mean shift criteria)

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Table 1. Optimal tolerances allocation using GA and PSO (a) Based on the worst case criteria

GA

PSO

Piston Cylinder Cost Time (s) 0.0162 0.0162 0.0037 0.0038 66.85 350 0.0013 0.0012 0.0005 0.0005

Piston Cylinder Min Ave Max Time(s) 0.0163 0.0163 0.0037 0.0037 66.74 66.74 66.74 83 0.0013 0.0013 0.0005 0.0005

(b) Based on the worst RSS criteria

GA

PSO

Piston Cylinder Cost Time (s) 0.0161 0.0161 0.0039 0.0038 65.92 330 0.0011 0.0012 0.0007 0.0006

Piston Cylinder Min Ave Max Time(s) 0.0161 0.0162 0.0039 0.0038 66.82 66.82 66.82 80 0.0011 0.0012 0.0007 0.0006

(c) Based on the worst Spotts' criteria

GA

PSO

Piston Cylinder Cost Time (s) 0.0160 0.0159 0.0038 0.0038 66.23 330 0.0012 0.0012 0.0006 0.0005

Piston Cylinder Min Ave Max Time(s) 0.0162 0.0162 0.0038 0.0038 65.93 65.93 65.93 78 0.0012 0.0012 0.0006 0.0006

(d) Based on the worst mean shift or Greenwood and Chase's unified criteria

GA

PSO

Piston Cylinder Cost Time (s) 0.0162 0.0151 0.0037 0.0038 66.26 350 0.0012 0.0011 0.0006 0.0006

Piston Cylinder Min Ave Max Time(s) 0.0161 0.0162 0.0039 0.0038 65.82 65.82 65.82 82 0.0011 0.0012 0.0006 0.0006

The optimal tolerance allocated using HPSO and GA based on the above four criteria and the corresponding CPU time are listed in Table 1. The computational results clearly indicate that HPSO outperformed GA in the terms of solution quality as well as computational expense. In addition, HPSO is able to find the optimum in each trial. It is necessary to point out that one important merit of PSO algorithm is the high precision of the solutions. However, due to the limitation of display capacity of the tables, the entire data are rounded. The statistical results obtained under the Greenwood and Chase’s estimated mean shift criteria are demonstrated in Figure.2. Similar curves can be obtained for other

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cases. Figure.2 reflects the general behavior about convergence of PSO algorithm. Sharply contrast with GA, the PSO algorithm has consistent convergence. The average and worst fitness are not fluctuant as in GA.

5 Conclusion Tolerance assignment, especially the simultaneous assignment, is very important in product design and machining. However, the optimization task is usually difficult to tackle due to the nonlinear, multi-variable and high constrained characteristics. In view of the memory characteristics of PSO, a new constraints handling strategy suit for PSO is designed. This new strategy can adequately utilize the historical information in PSO algorithm. The application on a cylinder-piston assembly example demonstrates its high efficiency and effectiveness. However, when we attempt to extend the proposed approach to the constrained optimization with large number of complex equality constraints, subtle drawbacks emerged, as the constrained range is so narrow that the equality constraints are hard to satisfy. This problem reveals the new research direction, i.e., the effective equality constraint handling strategy desirable to develop for PSO based nonlinear programming.

References 1. Ngoi, B.K.A., Teck, O.C.: A tolerancing optimization method for product design. Vol. 13. International Journal of Advanced Manufacturing Technology (1997) 290-299 2. Singh, P.K., Jain, P.K., Jain, S.C.: Simultaneous optimal selection of design and manufactur-ing tolerances with different stack-up conditions using genetic algorithms. Vol.41. Interna-tional Journal of Production Research (2003) 2411-2429 3. Gao, Y., Huang, M.: Optimal process tolerance balancing based on process capabilities. Vol.21. International Journal of Advanced Manufacturing Technology (2003) 501–507 4. Huang, M.F., Zhong, Y.R., Xu, Z.G.: Concurrent process tolerance design based on minimum product manufacturing cost and quality loss. Vol.25. International Journal of Advanced Manufacturing Technology (2004) 714-722 5. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neutral Networks, Perth, Australia (1995) 1942-1948 6. Shi, Y.H., Eberhart, R.C.: A modified particle swarm optimizer. In: Proceedings of IEEE Conference on Evolutionary Computation (1998) 69-73 7. Noorul Hap, A., Sivakumar, K., Saravanan, R., Karthikeyan, K.: Particle swarm optimization (PSO) algorithm for optimal machining allocation of clutch assembly. Vol.27. International Journal of Advanced Manufacturing Technology (2005) 865-869 8. Kennedy, J., Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufman, San Francisco (2001) 9. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. Proceedings of IEEE International Conference on Neutral Networks, Perth, Australia (1995) 1942-1948